AR(1) sequence with random coefficients: Regenerative properties and its application

Abstract: Let ${X_n}{n\ge0}$ be a sequence of real valued random variables such that $X_n=\rho_n X{n-1}+\epsilon_n,~n=1,2,\ldots$, where ${(\rho_n,\epsilon_n)}{n\ge1}$ are i.i.d. and independent of initial value (possibly random) $X_0$. In this paper it is shown that, under some natural conditions on the distribution of $(\rho_1,\epsilon_1)$, the sequence ${X_n}{n\ge0}$ is regenerative in the sense that it could be broken up into i.i.d. components. Further, when $\rho_1$ and $\epsilon_1$ are independent, we construct a non-parametric strongly consistent estimator of the characteristic functions of $\rho_1$ and $\epsilon_1$.